Explore BrainMass

Central Limit Theorem

     The central limit theorem (CLT) is one of the most useful theorems in statistics. It states that regardless of the population distribution, as long as all the variables are independently and identically distributed (i.i.d.), the sampling distribution of means will approach a normal distribution with mean equal to the population mean as the sampling size increases.

     Consider the population of Canada and imagine that you would like to find out what the mean height is. You start by taking a sample of size N and take the mean of that sample. But, this does not guarantee at all, that this will be the same as the population's mean. Now imagine that you continue to take more random samples of size N and find their means. Plotting all those means will give you a sampling distribution of means. The CLT states that this sampling distribution will approach a normal distribution as N grows large. Furthermore the mean of this sampling distribution will be approximately equal to the population mean as well. The sampling distribution variance will be approximately the population variance divided by N. The sampling standard deviation will be the square root of that. The central limit theorem evidently gives the ability to solve for the population mean with more accuracy than using one sample.  

Normal Distribution on a Portfolio

1) The daily returns on a portfolio are normally distributed with a mean of 0.001 and a standard deviation of 0.002. What is the probability that the average return for the portfolio over the next 100 days exceeds 0.0015? 2) In May 1983, after an extensive investigation by the Consumer Product Safety Commission, Honeywell

Statistical Analysis: Are we smarter than our parents

Read the article entitled, "Are We Smarter than Our Parents?" in chapter 5 of your textbook. This article addresses a study by Dr. James Flynn of the rise of the IQ rate over generations, and how statistics are involved in tracking this phenomenon, especially with reference to the material in chapter 5. Write a paper in which

Central Limit Theorem and Probabilities Under Standard Normal Curve

Will you check my work on 1, 2 & 5 and then help me with 3, 4 & 6... I don't know if the formula is still the same for all of them or different. 1. The heights of eighteen-year-old men are approximately normally distributed with a mean (µ) of 68 inches and a standard deviation (σ) of 3 inches. What is the probability that

Central Limit Theorem Explained in Layman's Terms

Could you please explain Central Limit Theorem to me in layman's terms and then read the article linked below and detail how it was used therein? Thank you.

The solution gives detailed steps on determining type I and type II errors in a specific hypothesis testing and using confidence interval method to make the conclusion of the same test. Next, central limit theorem is well defined and explained in details.

(b) The quality control engineer for a furniture manufacturer is interested in the mean amount of force necessary to produce cracks in stressed oak furniture. She performs a two-tailed test of the null hypothesis that the mean for the stressed oak furniture is 650. What are Type I and Type II errors for this problem? (c) A 95

Insurance Agents Nationwide in Ohio

Then the 27 listed insurance agents Nationwide Insurance in the metropolitan area of Toledo, Ohio. Calculate the average number of years that have worked in Nationwide. See attached. a) Select a random sample of four agents. If the Random numbers selected are: 02,59,51,25,14,29,77,69 and 18, then which dealers will be incl

Central Limit Theorem Conversation

Please see attachment. And please provide clear and thorough information so that I can understand it in future problems. Please provide assistance in the following 10 parts: 1. Boss has a question about a population: 2. To costly or can't do population so forced to sample 3. With sampling, will be sampling error (CLT


In the Department of Education at UR University, student records suggest that the population of students spends an average of 5.5 hrs per week playing organized sports. The population's standard deviation is 2.2 hrs per week. Based on a sample of 121 students, healthy lifestyles Inc. (HLI) would like to apply the central limit t

Normal Approximation to Binomial and Central Limit Theorem

Problems are also attached. 1. A fair die is rolled 25 times. Let X be the number of times a six is obtained. Find the exact value of P( X=6) and compare it with a normal approximation of P( X=6). 2. (Dice Sums). Suppose a fair die is rolled 1000 times. Compute an approximation to the probability that the sum of the 1000 r

Descriptive Statistics: Mean for E1 following a standard distribution

See attached file and please solve calculations in red. 2 Using the descriptive statistics data determined during Week One's weekly problem discussion, the mean for EI followed a standard distribution with a mean of 132.83 and a standard deviation of 15.68. If we selected another random sample of 50 participant

Statistics: 15 Problems

1. A population is normally distributed, with a mean of 23.45 and a standard deviation of 3.8. What is the probability of each of the following? a. Taking a sample of size 10 and obtaining a sample mean of 22 or more b. Taking a sample of size 4 and getting a sample mean of more than 26 2. The Statistical Abstract of the

Statistics: Normal Distribution, probabilities, values, central limit theorem

Included with each section or problem are reference examples and end of section exercises that can be used as a guide. Be sure to show your work in case partial credit is awarded. To receive full credit, work must be shown if applicable. Section 5.1: Introduction to Normal Distribution and the Standard Normal Distribution 1.

Distribution characteristics foundations

The normal distribution has several characteristics that make it the underlying foundation of much of inferential statistical tools. List these characteristics and explain how they contribute to the power of the distribution as foundation of inferential decision making.

Statistics discussion: Control chart distributions differences for a large sample

Please help with the following problem. Provide at least 200 words in the solution. Control charts are a popular (and often reasonable) way to statistically monitor a processes' performance. They are used for both variables (i.e. quantitative measures) and attributes (qualitative measures). Our authors state that the underly

Sampling Distribution/Central Limit Theorem

Sampling Distribution and the Central Limit Theorem Find the probabilities. a. From National Weather Service records, the annual snowfall in the TopKick Mountains has a mean of 92 inches and a standard deviation ? of 12 inches. If the snowfall from 25 randomly selected years are chosen, what it the probability that the

What is the probability that for a randomly selected customer the service time would exceed 3 minutes? If many samples of 64 were selected, what are mean and standard error of the mean expected to be? What is expected to be the shape of the distribution of sample means? If a random sample of 64 customers is selected, what is the probability that the sample mean would exceed 3 minutes?

The amount of time a bank teller spends with each customer has a population mean = 3.1 minutes and population standard deviation = 0.4 minute. a) What is the probability that for a randomly selected customer the service time would exceed 3 minutes? b) If many samples of 64 were selected, what are mean and standard error of t

Estimating Population Mean and Distribution

1) A state meat inspector in Iowa has been given the assignment of estimating the mean net weight of packages of ground chuck labeled "3 pounds". Of course he realizes that the weights cannot be precisely 3 pounds. A sample of 36 packages reveals the mean weight to be 3.01 pounds, with a standard deviation of 0.03 pounds. A) Wh

Sampling Methods and Central Limit Theorem (Please see the attachment)

(Please see the attachment) "Information from the American Institute of Insurance indicates the mean amount of life insurance per household in the United States is $110,000. This distribution follows the normal distribution with a standard deviation of $40,000. " "a. If we select a random sample of 50 hou

Normal Distribution and Central Limit Theorem

The mean amount purchased by a typical customer at Churchill's Grocery Store is $23.50 with a standard deviation of $5.00. Assume the distribution of amounts purchased follows the normal distribution. For a sample of 50 customers, answer the following questions. a. What is the likelihood the sample mean is at least $25.00? b

Understanding the Central Limit Theorem.

Visit the following Web site Central Limit Theorem Applet and read what is posted: You will choose from the pull down menu at the bottom of the page both the number of dice and the number of rolls at a time. When you "click" you will be virtually rolling your dice. Complete t

Sampling distibutions, central limit theorem, and probabilities

4. A computer supply house receives a large shipment of floppy disks each week. Past experience has shown that the number of flaws per disk can be described by the following probability distribution: Number of Flaws per Floppy Disk Probability 0 .65 1 .2 2

Quantative Research Methods: Sample Size, Confidence Interval

Question 1: As a sample size approaches infinity, how does the student's t distribution compare to the normal z distribution? When a researcher draws a sample from a normal distribution, what can one conclude about the sample distribution? Explain. Question 2: A mayoral election race is tightly contested. In a random sam

Discussing the Main Points of the Central Limit Theorem for a Mean

Question: Why is population shape of concern when estimating a mean? What does sample size have to do with it? Scenario 1: A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were as given below: 3.087 3.131 3.241 3.241 3.270 3.353 3.4

Two questions on Central Limit Theorem - Confidence Level and Sample Size

#1 Answer the following based upon the implications of the Central Limit Theorem (assume sample size larger than 30 when samples are mentioned in (a) and (b) below). a) How does the mean of the sampling distribution of all possible sample means from a population compare to the mean of the population? b) How does the stan

Statistics Definitions

What is the sampling distribution of sample means? What is the mean of the sampling distribution of sample means? What is its standard deviation? How is that standard deviation affected by the sample size? What does the central limit theorem state about that distribution?

Normal Distribution Using Central Limit Theorem

The mean weight of newborn babies in a Long Island Community is 7.5 lbs with a standard deviation of 1.4 lbs. What is the probability that a random sample of 49 babies has a sample mean weight of at least 7.2lbs?